topology and exotic orders in quantum solids

Topology and exotic orders in quantum solids

Ying Ran Boston College

ITP, CAS, June 2013

This talk is about:

• Zoology of topological quantum phases in solids Introduction and overview.

• How to realize them in materials? where to look for them? what kind of new materials?

• How to systematically understand them? New theoretical framework?

The “Standard Model” of condensed matter

• Different phases are characterized by different symmetries.

• Emergent Laudau order parameter

• Successfully describes a large set of phenomena in solids

Landau’s Fermi Liquid (metals)

Landau Theory of broken symmetry.

First topological phases: IQHE and FQHE• In 1980’s, integer/fractional quantum hall phases

2D electron gas in a strong magnetic field

--- Quantized Hall conductance:

von Klitzing, Tsui, Stormer, Laughlin ….

First topological phases: IQHE and FQHE• In 1980’s, integer/fractional quantum hall phases --- striking counterexamples of the “Standard Model”: All have the same symmetry, yet there are many different phases!

2D electron gas in a strong magnetic field

--- Quantized Hall conductance:

von Klitzing, Tsui, Stormer, Laughlin ….

Beyond the “Standard Model” in solids?• Previously, violations only in “extreme conditions” one dimension, 2DEG in strong magnetic field


• New patterns of emergence in solidse.g.

• Topological insulators • Quantum spin liquids

Bi2Se3HgTe quantum wellHerbertsmithitedmit organic salts


• New patterns of emergence in solidse.g.

• Topological insulators

• Topological superconductors

• Quantum spin liquids

• Fractional Chern insulators --fractional quantum hall states in solids in the absence of magnetic field

So far not realized in experiments

Bi2Se3HgTe quantum wellHerbertsmithitedmit organic salts

??

• With a growing list of topological phases, it may be helpful to organize them in a certain way --- a zoology.


• In fact, all topological quantum phases can be viewed as:

• Generalizations of integer quantum hall phases

• Generalizations of fractional quantum hall phases


• In fact, all topological quantum phases can be viewed as:

• To perform generalization, helpful to review the key features of integer/fractional quantum hall phases

--- Why we call them topological phases?

• Generalizations of integer quantum hall phases

• Generalizations of fractional quantum hall phases

Integer quantum hall phases

QuantumMechanics

E

Landau Levels

2DEG in a magnetic field

Integer quantum hall phases

QuantumMechanics

E

Landau Levels


EF

Integer quantum hall phases: key features

• Landau levels are energy bands with non-trivial topology: Chern number C =1 Thouless-Kohmoto-Nightingale-den Nijs (1982)

Chern number = Integral of Berry’s curvatures of wavefunctions

QuantumMechanics

E

Landau Levels


EF

C=1



Chern number = Integral of Berry’s curvatures of wavefunctions

Analogy: genus g (number of handles). Integral of Gaussian curvature: K

QuantumMechanics

E

Landau Levels


EF

)1(4 gdSK

g=0 g=1

C=1

from Charlie Kane’s website



• IQH phases are band insulators: ordinary gapped bulk excitations

QuantumMechanics

E

Landau Levels


EF

C=1

Band insulator




• Characteristic gapless edge modes

QuantumMechanics

E

Landau Levels


EF

C=1




• Characteristic gapless edge modes

QuantumMechanics

E

Landau Levels


EF

C=1

--- Similar features in generalized phases

Generalized “integer phases”Examples:

• Topological insulators in spin-orbit coupled solids (Kane, Mele, Zhang, Bernevig, Molenkamp, Hasan, Fu, Qi, Roy Balents, Moore, Vanderbilt…… )

2D TI: HgTe quantum well 3D TI: Bi2Se3, Bi2Te3,….



Key features:

(1) Band insulator --- ordinary gapped bulk excitations

A schematic band structure

Gap



Key features:


(2) Bands with nontrivial topology: Z2 index (0 or 1 instead of integer)

A schematic band structure

Gap



Key features:



(3) Characteristic gapless edge modes



Key features:




(2), (3) protected by time-reversal symmetry ---They are gone if time-reversal symmetry is broken. (e.g., magnetic impurities)

Generalized “integer phases”

Key features:





Examples:


• Other examples: topological superconductors, bosonic analogs ….

Symmetry protected topological phases

Key features:





Examples:


• Other examples: topological superconductors, bosonic analogs ….

The modern view of gapped quantum phases

Landau phases

….

Isingferromagnet

Isingparamagnet

“Standard model”

+ Topological Phases


Landau phases

….

Isingferromagnet

Isingparamagnet



Generalization of FQH phases

Generalization of IQH phases


Landau phases

….

Isingferromagnet

Isingparamagnet



symmetry protected topological phases

Top. Insulator

Top. superconductor

….

• Ordinary bulk excitation• Symmetry-protected gapless edge modes



Landau phases

….

Isingferromagnet

Isingparamagnet




Top. Insulator

Top. superconductor

….



Key features?

Fractional quantum hall phases

QuantumMechanics

E

Landau Levels

EF

C=1

Integer plateaus


• A partially filled Laudau level: C=1 flat band

QuantumMechanics

E

Landau Levels

EF

C=1

?


• A partially filled Laudau level: C=1 flat band

• Electron-electron Coulomb interactions lift degeneracy Fractional quantum hall phases

QuantumMechanics

E

Landau Levels

EF

C=1

fractional plateaus

Fractional quantum hall phases: Key featuresApart from the quantized hall conductance

• NOT a band insulator in the bulk anyon excitations with a finite gap:

Fractional statistics

Fractional quantum hall phases: Key featuresApart from the quantized hall conductance


• Topological ground state degeneracy

sphere

E

Gap

V.S.

E

Gap

torus


Wen,Niu 1990

Fractional quantum hall phases: Key featuresRobust towards any local perturbations! DO NOT require symmetry



sphere

E

Gap

V.S.

E

Gap

torus


• Wavefunctions locally identical• Local perturbations cannot lift degeneracy

Fractional quantum hall phases: Key featuresRobust towards any local perturbations! DO NOT require symmetry Protected by long-range quantum entanglement• NOT a band insulator in the bulk anyon excitations with a finite gap:


sphere

E

Gap

V.S.

E

Gap

torus


• Wavefunctions locally identical• Local perturbations cannot lift degeneracy

Fractional quantum hall phases: Key featuresRobust towards any local perturbations! DO NOT require symmetry Protected by long-range quantum entanglement• NOT a band insulator in the bulk anyon excitations with a finite gap:


sphere

E

Gap

V.S.

E

Gap

torus


These features can be used to characterize different phases.

Generalized “Fractional phases”

Generalized “Fractional phases”Examples:• Gapped quantum spin liquids

-- Mott insulators without any symmetry breaking

Candidate material Herbertsmithite:Gapless or a small gap?(Helton,Lee,McQueen,Nocera,Broholm,….)



Hastings’ Theorem (2004):A gapped quantum spin liquid has ground state deg. on torus.

E

Gap

torus



Hastings’ Theorem (2004):A gapped quantum spin liquid has ground state deg. on torus.

But by definition of QSL, not due to symmetry breakingdue to long-range entanglement

E

Gap

torus



shared key features:protected by long-range quantum entanglement, do not require any symmetry


• anyon excitations • Topological ground state deg.



shared key features: can be used to characterize different phasesprotected by long-range quantum entanglement, do not require any symmetry



entanglement protected topological phasesExamples:• Gapped quantum spin liquids


shared key features: can be used to characterize different phasesprotected by long-range quantum entanglement, do not require any symmetry




Landau phases

….

Isingferromagnet

Isingparamagnet




Top. Insulator

Top. superconductor

….




Landau phases

….

Isingferromagnet

Isingparamagnet




Top. Insulator

Top. superconductor

….


entanglement protected topological phases

Gapped spin

liquid

Fractional Chern

insulator ….

• Anyon bulk excitation• Topological ground state degeneracy• Robust even without any symmetry


Landau phases

….

Isingferromagnet

Isingparamagnet




Top. Insulator

Top. superconductor

….



Gapped spin

liquid

Fractional Chern

insulator ….

Will come back to this later


This talk is about:




This talk is about:



Searching for topological phases in transition metal oxide heterostructures

Xiao,Zhu,YR,Nagaosa,Okamoto, Nat. Commun. (2011) Yang,Zhu,Xiao,Okamoto,Wang,YR, PRB Rapid Commun. (2011) Wang, YR, PRB Rapid Commun. (2011)

Motivation• A growing family of topological insulators:

‣ CdHgTe/HgTe/CdHgTe (Bernevig et al, Science 2006, Konig et al, Science2007)

‣ Bi1-xSbx (Fu and Kane, PRB 2007, Hsieh et al, Nature 2008) ‣ Bi2Se3, Bi2Te3, Sb2Te3 (Zhang et al, Nat Phys 2009, Xia et al, Nat Phys

2009, Chen et al, Science 2009) ‣ TlBiTe2 and TlBiSe2 (Lin et al, PRL 2010, Yan et al, EPL 2010, Sato et al,

PRL 2010, Chen et al, PRL 2011) ‣ Half-heuslers, Chalcopyrites (Lin et al, Nat Mat. 2010, Chadov et al, Nat Mat

2010, Xiao et al, PRL, 2010, Feng et al, PRL 2010) ‣ Many more...

Motivation• A growing family of topological insulators:

‣ CdHgTe/HgTe/CdHgTe (Bernevig et al, Science 2006, Konig et al, Science2007)

‣ Bi1-xSbx (Fu and Kane, PRB 2007, Hsieh et al, Nature 2008) ‣ Bi2Se3, Bi2Te3, Sb2Te3 (Zhang et al, Nat Phys 2009, Xia et al, Nat Phys

2009, Chen et al, Science 2009) ‣ TlBiTe2 and TlBiSe2 (Lin et al, PRL 2010, Yan et al, EPL 2010, Sato et al,

PRL 2010, Chen et al, PRL 2011) ‣ Half-heuslers, Chalcopyrites (Lin et al, Nat Mat. 2010, Chadov et al, Nat Mat

2010, Xiao et al, PRL, 2010, Feng et al, PRL 2010) ‣ Many more...

---they are all s/p-orbital electronic systems

Motivation• What about d-orbital?


Correlation-driven physics:e.g., various symmetry breaking phases• Superconductivity• Magnetism• Ferroelectricity

….


Correlation-driven physics:e.g., various symmetry breaking phases• Superconductivity• Magnetism• Ferroelectricity

….

+ TI physics ?


Correlation-driven physics:e.g., various symmetry breaking states• Superconductivity• Magnetism• Ferroelectricity

….

+ TI physics ?

• Novel applications of TI physics require proximity effects between TIs and symmetry-breaking states.

(e.g., magnetoelectric effects, Majorana fermions)

• New regime: interplay between Mott physics and TI physics

I will show:Certain transition metal oxide heterostructures could host:• room-temperature 2D TI phases

I will show:Certain transition metal oxide heterostructures could host:• room-temperature 2D TI phases• and much more than that: quantum anomalous hall insulator, abelian/non-abelian fractional Chern insulators, Dirac half-semimetal, quantum spin liquids……

I will show:Certain transition metal oxide heterostructures could host:• room-temperature 2D TI phases• and much more than that: quantum anomalous hall insulator, abelian/non-abelian fractional Chern insulators, Dirac half-semimetal, quantum spin liquids……

Lesson from previous TI materials (HgTe, Bi2Se3…): semi-metal + spin-orbit interaction: generates (inverts) the band gap.

E

k

EF

+Spin-orbit coupling

E

k

EFGap

Heterostructures of transition metal oxides

• Layered structure can be prepared with atomic precision

• Great flexibility: tunable lattice constant, carrier concentration, spin-orbit interaction, correlation strength...

• Correlation physics of d-orbitals: Mott physics, magnetism, superconductivity…

Crystal structure• Current technology focus on perovskites ABO3.

• Experimental efforts are mainly on interface/hetero-structures grown along the (001) direction

For example, superconductivity is found on STO/LAO interface

Perovskite structure of SrTiO3Reyren et al, Science 2007

• Previously, possible topological phases have not been investigated in TMOH.

• This is partially because the current efforts are on (001) direction. (square lattice---large fermi surface, or large band gap…)

• I will show that, heterostructures grown along the (111) direction are particularly interesting for topological phases of matter.

Z

Square lattice of transition metal atomsX

Y

X

Y

Perovskite (111)-bilayer

• d-electrons hopping on a buckled honeycomb lattice

Example:

LaAlO3substrate

LaAlO3substrate

LaAuO3

(111) direction



Example:

LaAlO3substrate

LaAlO3substrate

LaAuO3

(111) direction

Graphene-like band structure?



• Naturally give semi-metallic band structure --Similar physics to graphene? (“correlated versions” of graphene ? )

Example:

LaAlO3substrate

LaAlO3substrate

LaAuO3

(111) direction

d-electrons in a crystal

d-orbitals5x2=10 states

OctahedralCrystal field

t2g

eg

d-electrons in a crystal

d-orbitals5x2=10 states

OctahedralCrystal field

t2g

eg

Example

Au 3+:

8 electrons in 5d orbitals

eg orbitals half-filled

Example: LaAlO3/LaAuO3/LaAlO3 (111) bilayer

• Tight-binding band structure without Spin-Orbital coupling

Xiao,Zhu,YR et.al, (2011)

EF

• Interestingly, similar to graphene + 2 flat bands.

The exact flatness of these bands are consequence of the nearest neighbor model.Further neighbor hoppings destroy the exact flatness.


• Tight-binding band structure without Spin-Orbital coupling


EF

• Interestingly, similar to graphene + 2 flat bands.

• Can S-O coupling generate topological gap? similar to graphene (Kane&Mele 2005)…


• Tight-binding band structure with Spin-Orbital coupling


EF




• LaAlO3/LaAuO3/LaAlO3 (111) bilayer is a 2D TI.

EF EF

with S-O coupling:Gapless edge states




• LaAlO3/LaAuO3/LaAlO3 (111) bilayer is a 2D TI.

• Topological band is nearly flat!

EF EF


• Comparing with first principle calculation:


• LaAlO3/LaAuO3/LaAlO3 (111) bilayer is a room-temp. 2D TI.

Tight-binding analysis First-principle calculation


• Comparing with first principle calculation:


• LaAlO3/LaAuO3/LaAlO3 (111) bilayer is a room-temp. 2D TI.

• Flat band is slightly dispersive (further neighbor hopping)

Tight-binding analysis First-principle calculation

What if the nearly flat band is partially filled?

What if the nearly flat band is partially filled? • Lesson from FQHE:

Partially filled topological flat band (Laudau level) + Correlation:

Fractional topological phases

E

Landau Levels

EF

C=1

fractional plateaus

What if the nearly flat band is partially filled? • Fractional topological phases?

---Realizable: e.g., electron-doped SrTiO3/SrPtO3/SrTiO3

EF

+ Correlation


What if the nearly flat band is partially filled? • Fractional topological phases?

---Realizable: e.g., electron-doped SrTiO3/SrPtO3/SrTiO3

• Our calculations show signature of fractional quantum hall effects!

--- FQHE in the absence of magnetic field, has been called “fractional Chern insulator” (Mudry, Chamon,Tang, Wen, Sun, Sheng, Gu,Bernevig, Fiete, 2011….)

EF

+ Correlation


What if the nearly flat band is partially filled? • Our calculations:

EF


+ Correlation


EF

Ferromagnetism

Correlation

EF

nearly flat band with C=1--- analogy of Landau levelIn the absence of mag. field.



With realistic interactions…

EF


Correlation

EF

And choose band-filling º=1/3

Ferromagnetism


Exact diagonalization simulations show:

EF


Correlation

EF

Ferromagnetism

(1) 3-fold ground state degeneracy on torus (2) Quantized hall conductance:

(many-body Chern number =1/3)


Numerical signatures of 1/3-Laughlin fractional Chern insulator:

EF


Correlation

EF

Ferromagnetism

(1) 3-fold ground state degeneracy on torus (2) Quantized hall conductance:

(many-body Chern number =1/3)

Why fractional Chern insulators are interesting?

Why fractional Chern insulators are interesting?• For practical purpose: High-temperature FQHE without magnetic field


• For fundamental science’s purpose:Are there intrinsically new regime/phases?


• For fundamental science’s purpose:Are there intrinsically new regime/phases?Yes:• A band structure can have Chern number C >1 bands --- no analog in Landau Level (C=1)

• Intrinsically new regime: Partially filled nearly flat C>1 bands + correlation Lu,YR(2011)



• Intrinsically new regime: Partially filled nearly flat C>1 bands + correlation

e.g., the natural counterpart of 1/3-Laughlin state in C=2 band is non-Abelian --- SU(3)1 Abelian Chern-Simons theory SU(3)2 non-Abelian Chern-Simons theory

Lu,YR(2011)



• Intrinsically new regime: Partially filled nearly flat C>1 bands + correlation

• Is it possible to realize nearly-flat C>1 bands in materials?

Lu,YR(2011)

Nearly flat C=2 bands: SrIrO3 (111) trilayer• Trilayer: transition metal atoms form a Dice-lattice. Wang,YR (2011)


Dice lattice is known to support flat band. (one of the many Lieb’s theorems)

Without S-O With S-O



The spin degenerate flat band is half-filled. --- correlation-driven ferromagnetism




The spin degenerate flat band is half-filled. --- correlation-driven ferromagnetism


+ correlation

Our calculations show: ferromagnetism nearly-flat C=2 bands

What about experiments?

What about experiments?• Motivated by our theoretical investigations on possible

correlation-driven topological phases in LaNiO3 (111) bilayer:

---Strongly correlated 3d electrons on honeycomb lattice Within realistic regime, we identified:• Dirac half-semimetal (spinless graphene)• Quantum anomalous hall insulator

Yang,YR,et.al (2011)Also Fiete et.al, (2011)

Experiment progress!• Motivated by our theoretical investigations on possible

correlation-driven topological phases in LaNiO3 (111) bilayer:

• The successful synthesis of (111) bilayer LaAlO3/LaNiO3/LaAlO3 heterostructure was reported recently:

Yang,YR,et.al (2011)Also Fiete et.al, (2011)

This talk is about:




Lu,YR, PRB (2012)Mesaros,YR (arXiv. Dec. 2012, to appear on PRB 2013)

Motivation• Crystals: lessons from the “standard model”

Crystals = spontaneous breaking of translational symmetry • symmetry group theory allows systematic understandings:

230 space groupsAll realized in nature!

Liquid

Cool down

Crystal

Motivation• Crystals: lessons from the “standard model”

Crystals = spontaneous breaking of translational symmetry • symmetry group theory allows systematic understandings:

• What is the “group theory” for topological phases?

230 space groupsAll realized in nature!

Liquid

Cool down

Crystal


Landau phases

….

Isingferromagnet

Isingparamagnet




Top. Insulator

Top. superconductor

….



Gapped spin

liquid

Fractional Chern

insulator ….



Landau phases

….

Group theory




Group cohomology K-theory

….



Gauge theory

Tensor Category ….


New mathematics introducedfor systematic understandings

(Wen, Kitaev, Levin, Senthil, Turner, Pollmann,Chen, Gu, Vishwanath, Lu, Ryu, Schnyder, Ludwig….)




….



Gauge theory



• What about topological phases protected by both symmetry AND entanglement?




….



Gauge theory




• Directly relevant to physical systems

sym. and entanglement show up together:

e.g.:Quantum spin liquid --- spin rotation sym.

Fractional Chern insulator --- lattice sym.Lu, YR, PRB (2012)




….



Gauge theory




• Directly relevant to physical systems

sym. and entanglement show up together:

e.g.:Quantum spin liquid --- spin rotation sym.

Fractional Chern insulator --- lattice sym.

• How to glue the two pieces together?

Lu, YR, PRB (2012)

Roughly speaking, my picture is like:• The space of topological phases:

Group

cohomology

K-theory….

•Ordinary bulk excitation

•Symmetry-protected gapless edge modes


Gauge theory

Tensor category

….




Group

cohomology

K-theory….

•Ordinary bulk excitation

•Symmetry-protected gapless edge modes


Gauge theory

Tensor category

….



sym. but no entanglemententanglement but no sym.



TOPOLOGICAL PHASES PROTECTEDBY BOTH SYMMETRY AND ENTANGLEMENT


?

sym. but no entanglemententanglement but no sym.

Roughly speaking, my picture is like:• The space of topological phases: Systematic understanding in the “bulk” --- Mission impossible?




?

Roughly speaking, my picture is like:• The space of topological phases: Systematic understanding in the “bulk” --- Mission impossible?

--- at least we can provide answers to a certain level




?

A classification of topological phasesprotected by both symmetry and entanglement

• Assumptions:(1) bosonic gapped quantum phases (e.g. quantum spin systems)(2) A local symmetry group SG. (e.g. spin rotations)(3) Entanglement described by a gauge group GG

Mesaros, YR (2012)



In d-spatial dimension, topological phases protected by sym. SG and entanglement GG are classified by the (d+1)th cohomology group:

Hd+1(SG£GG, U(1))

Mesaros, YR (2012)




Hd+1(SG£GG, U(1))

For example, when SG=Z2 (Ising symmetry) and GG=Z2,

H3(Z2£ Z2, U(1))=Z23

---in 2-spatial dimension, 8 Ising paramagnetic phases whose entanglement described by Z2 gauge group.

Mesaros, YR (2012)




Hd+1(SG£GG, U(1))

And we provide exactly solvable models realizing each phase in the classification.

Mesaros, YR (2012)

H3(SG£GG, U(1)) = H3(SG, U(1)) £ H3(GG, U(1)) £ H2(SG, GG) £ ….

A math theoremKünneth formula:


A math theorem

Now has full physical meaning

Künneth formula:

SG: symmetryGG: entanglement

New understanding in the “bulk”





Mesaros, YR (2012)






Chen, et.al, 2011

Mesaros, YR (2012)






Chen, et.al, 2011Dijkgraaf-Witten, 1990

Mesaros, YR (2012)






Chen, et.al, 2011Dijkgraaf-Witten, 1990

Mesaros, YR (2012)

New results: Characterizing different interplays between symmetry and entanglement

Example: new phases• 2-spatial dimension:

SG=Z2 (Ising symmetry), GG=Z2 £ Z2

H3(SG£ GG, U(1) ) = Z27 ---128 phases.

Mesaros, YR (2012)

But H3(SG,U(1))=Z2 H3(GG,U(1))=Z23 There are Z2

3 new indices in the “bulk”



H3(SG£ GG, U(1) ) = Z27 ---128 phases.

• Among them, we identify phases hosting new kinds of interplay between symmetry and entanglement.

Consequence: new phenomenaFor excited state with two quasiparticles

Mesaros, YR (2012)

E

Ising sym. protect two-fold degeneracy!

E

degeneracy liftedif sym. is broken



H3(SG£ GG, U(1) ) = Z27 ---128 phases.

• Among them, we identify phases hosting new kinds of interplay between symmetry and entanglement.

Consequence: new phenomenaFor excited state with two quasiparticles

Mesaros, YR (2012)

E

Ising sym. protect two-fold degeneracy!

E

degeneracy liftedif sym. is broken

In order to obtain results like this ….

new kinds of calculations …• Solving models in a geometric fashion:

Models:

Solution of models:

Mesaros, YR (2012)

Summary• Topological quantum phases are beyond the “standard model”.

• There are many different kinds of them. Some have been realized.

Experimentally:• Progress on new materials would be essential. --- e.g. searching for topological phases in transition metal oxide heterostructures

Theoretically:• introducing new framework, new methods --- e.g. systematic understanding of topological phases protected by both sym. and entanglement


Lu, YR, PRB (2012) Mesaros, YR (arXiv Dec. 2012, to appear on PRB 2013)

Summary• Topological quantum phases are beyond the “standard model”.

• There are many different kinds of them. Some have been realized.

Experimentally:• Progress on new materials would be essential. --- e.g. searching for topological phases in transition metal oxide heterostructures

Theoretically:• introducing new framework, new methods --- e.g. systematic understanding of topological phases protected by both sym. and entanglement

In addition, finding new detectable signatures and developing new experimental probes are also very important.


Lu, YR, PRB (2012) Mesaros, YR (arXiv Dec. 2012, to appear on PRB 2013)

Acknowledgement• Oak Ridge National Lab: Di Xiao(CMU), Satoshi Okamoto,

Wenguang Zhu

• MIT: Fa Wang ( PKU)

• Tokyo Univ.: Naoto Nagaosa

• Boston College: Yuan-Ming Lu (UC Berkeley), Bing Ye, Kaiyu Yang, Andrej Mesaros, Ziqiang Wang

Thank you!

topology and exotic orders in quantum solids

Documents

quantum solidsying

fractional quantum hall

quantized hall conductance

absence of magnetic

solids introduction

kind of new materials

new theoretical framework

extreme conditions